MATH 213 A – Discrete Mathematics for Computer Science

Dr. (Mr.) Bancroft

The inhabitants of the island created by Smullyan are peculiar. They consist of knights and knaves. Knights always tell the truth and knaves always lie. You encounter two people A and B. Determine, if possible, what A and B are (either a knight or a knave) from the way they address you.

A says “I am a knave or B is a knight.”

B says nothing.

1.1 Logic

Logic-

Proposition-

• Notation:

• Negation:

Truth Tables

Conjunction of p and q:

Disjunction of p and q:

Exclusive or:

Implication/Conditional:

Biconditional:

Operations on Implications:

Converse:

Contrapositive:

Inverse:

More complicated truth tables

Logic and Bit Operators

1.2 Propositional Equivalences (Several Definitions):

Compound proposition-

Tautology-

Contradiction-

Contingency-

Logical Equivalence

Using Truth Tables to Demonstrate Logical Equivalence

Show that and are logically equivalent.

Some Commonly used Logical Equivalences

Other Commonly used Logical Equivalences

De Morgan’s Laws

Let’s revisit the knight and knave problem:

A says “I am a knave or B is a knight.”B says nothing.

Arguments using logical equivalence“Chain” of equivalences (recall the way you proved trig identities)Examples:

1. Prove is a tautology.

2. Show that and are logically equivalent (again), this time using equivalences from the tables.

Using a Computer to Find Tautologies

Practical only with small numbers of propositional variables.

How many rows does the truth table contain for a compound proposition containing 3 variables?

5 variables?

10 variables?

100 variables?

1.3 – Predicates and Quantifiers

Is “” a proposition?

Predicates, or Propositional functions

Note that if x has no meaning, then P(x) is just a form.

The domain of x is …

There are two ways to give meaning to a predicate P(x):

The Universal QuantifierThe universal quantification of the predicate P(x) is the proposition which states that…

In symbols,

Example: (Let the domain of discourse be all real numbers)

The Existential QuantifierThe existential quantification of the predicate P(x) is the proposition which states that…

In symbols,

Example: (Let the universe of discourse be all people)

Looping to Determine the Truth of a Quantified Statement

Free and Bound Variables

“Scope” of a quantifier

Relationship with Conjunction and Disjunction

Negating a Quantified Statement

Translating into English Sentences

P(x) = “x likes to fly kites”Q(x,y) = “x knows y”

))(),(( xPxJoanQx

L(x,y) = “x likes y”

)),(),(( CalvinxLxSusieLx

Translating from English Sentences“All cats are gray”

“There are pigs which can fly”

Logic Programming

sibling(X,Y) :- parent(Z,X), parent(Z,Y), X \= Y.brother(X,Y) :- sibling(X,Y), male(X).sister(X,Y) :- sibling(X,Y), female(X).male(chris).male(mark).female(anne).female(erin).female(jessica).female(tracy).parent(chris,mark).parent(anne,mark).parent(chris,erin).parent(anne,erin).parent(chris,jessica).parent(anne,jessica).parent(chris,tracy).parent(anne,tracy).

?sibling(erin,jessica)?sibling(mark,chris)

?parent(Z,tracy)

Section 1.4 – Nested Quantifiers

)())1(0(

yyxyxxyyxx

Examples:

Order of quantification matters!Example: M(x,y) = “x is y’s mother”

),(),(yxyMxyxxMy

Another Example

)),()((

)),()((

yxMySxy

yxxMySy

Translate each of these, where M is as above and S(x) = “x is a student” …

English to First-Order LogicLet L(x,y) = “x loves y”. Translate…

“Everybody loves somebody.”

“There are people who love everybody”

“All students love each other”